

we use the regularity theory for elliptic problems with bulk-surface coupling (see [1]}) to derive that
\(\begin{aligned}\Vert  \phi  \Vert _{H^2(\Omega )}^2&\le \Vert  (\phi ,\psi ) \Vert _{\mathcal {H}^2}^2\le C \Vert  (-\Delta \phi , -\Delta _\Gamma \psi + \partial _\mathbf {n}\phi ) \Vert _{\mathcal {L}^2}^2\\& \le C \Big ( \Vert  \Delta \phi  \Vert _{L^2(\Omega )}^2+ \Vert  \theta  \Vert _{L^2(\Gamma )}^2+ \big \Vert  G^{\prime }(\psi _1) - G^{\prime }(\psi _2) \big \Vert _{L^2(\Gamma )}^2 \Big )\\&\le C_0 \left( \Vert  \Delta \phi  \Vert _{L^2(\Omega )}^2 + \alpha ^2 \Vert  \theta  \Vert _{L^2(\Gamma )}^2 \right)+ \frac{C}{\alpha ^2} \Vert  \psi  \Vert _{L^2(\Gamma )}^2.\end{aligned}\) 
